Ellipticized lens providing balanced astigmatism

ABSTRACT

An ellipticized singlet azimuth versus elevation optimized and aperture  eemized nonspherical lens antenna of very low or even minimal F-number providing balanced astigmatism for wide angle acoustic, microwave or optical applications is described. The lens has an elliptical periphery and surfaces defined by a system of nonlinear partial differential equations, the surfaces acting together to produce two perfect primary off-axis foci F and F&#39; at a finite distance in back of the lens and two perfect conjugate off-axis foci F.sub.∞ and F&#39;.sub.∞ in front of the lens at infinity; i.e., the lens simultaneously focuses energy from the primary foci F and F&#39; into two off-axis parallel ray plane wave beams directed towards infinity at equal but opposite angles with respect to the lens axis. The lens may be built of various materials depending on its intended application in acoustics, microwaves or optics.

STATEMENT OF GOVERNMENT INTEREST

The invention described herein may be manufactured and used by or forthe Government of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

BACKGROUND OF THE INVENTION

In the design of singlet microwave dielectric lenses in theelectromagnetic spectrum and acoustic lenses for underwater soundapplications or for certain applications in the optical spectrum inwhich wide angle scanning or fixed off-axis operation is required, it isfrequently desired to achieve greater off-axis scan in one plane than inthe other, wide angle azimuth performance usually being more necessarythan elevation scan. Due to requirements for weight minimization andcompact packaging for aircraft and submersible vehicle installationswhere both system volume and weight are at a premium, it is furthergenerally desirable to extremize the lens in the sense of maximizing thelens aperture or minimizing the lens volume and sometimes simultaneouslyto achieve a minimum, or at any rate a near minimum, F-number.

On the other hand, in both the microwave and acoustic applications forthe most part, excluding only those areas of medical acousticradiography and related fields in which frequencies higher than 10megahertz are commonly used, the wavelengths encountered are of theorder of a millimeter, a centimeter, or a decimeter so that there is noneed to restrict the lens design to spherical surfaces grindable only byself-correcting motions as in optical lens manufacture, but rather,nonspherical lenses of quite complex shape may be readily used. Withthis relaxation of the design requirements, consideration in radar, andmore recently in acoustics, was first given to nonspherical butrotationally invariant lenses, i.e., nonspherical lenses whose surfacesare surfaces of revolution about the lens axis such as the aplanatic,generalized aplanatic and bifocal or Doppler lenses of L. C. Martin, F.G. Friedlander and R. L. Sternberg. Such lenses, however, like anyordinary singlet lens, spherical or nonspherical, invariably suffer whenextremized from astigmatism at off-axis points with the magnitude ofthat aberration increasing with increasing scan angle and decreasingF-number.

SUMMARY OF THE INVENTION

In order to circumvent the astigmatic defect of all single elementrotationally invariant extremized lenses at off-axis angles for thepurpose of achieving increased scan in the azimuth plane at theanticipated sacrifice of a reduction in elevation scanning, thepossibility of designing a geometrically perfect nonspherical bifocal orDoppler lens in three-space, symmetric with respect to two orthogonalplanes through the lens axis, one of which contains its design foci, andcomplete with minimal lens volume and a very low, or in some instancesan even minimal, F-number is considered. Such a nonspherical lens, hasaxial symmetry about the z-axis, and also plane symmetry about both thexz and yz-planes, but no surfaces of revolution and has balancedastigmatism; i.e., negative astigmatism on axis, zero astigmatism at itsoff-axis design points or primary foci F and F' and positive astigmatismat still further off-axis points and, consequently, offers theopportunity for an optimized trade off between azimuth and elevationscanning capability by suitable optimum choice of the basic lensparameters.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates the lens surfaces S and S' and their refractiveaffect on the ray R and R_(o) ' from the focus F to the wavefront W;

FIG. 2 illustrates the meridional plane cross-section of the lens ofFIG. 1 and the rays R_(N).sbsb.1 to R_(N).sbsb.N and R'_(N).sbsb.1 toR'_(N).sbsb.N used in calculation of the eigenvalue like angles φ₁ andφ'₁ ;

FIG. 3 illustrates meridional plane cross-section of the lens of FIG. 1showing the particular case of FIG. 2, where N=1; and

FIGS. 4A and 4B illustrate the ray tracing results of successively moreaccurate determinations of the lens of FIG. 1.

DESCRIPTION OF THE PREFERRED EMBODIMENT

The geometrical shape of the ellipticized azimuth versus elevationoptimized and aperture extremized nonspherical singlet lens comprisingthe present invention is characterized by the following boundary valueproblem and is an extension of the inventor's prior lens antennainvention as described in U.S. Pat. No. 3,146,451.

Referring now to FIG. 1 there is shown a lens with a graphicalrepresentation of a general ray structure. The lens may be built ofvarious materials depending on its intended application in accoustics,microwaves or optics. A dual ray structure is also described but is notshown in FIG. 1 as it would excessively complicate the drawing and canbe readily understood from the description.

Given the focal distance |z_(o) |, the lens diameter 2b_(o), the indexof refraction of the lens material n_(o) >1, the offaxis design angleψ_(o) and the wavelength λ_(o) of the incident electromagnetic oracoustic wave to be focused by the lens, i.e., given the parameters

    |z.sub.o |,2b.sub.o,n.sub.o >1,ψ.sub.o,λ.sub.o,                            (1)

let the lens surfaces S and S' in FIG. 1 have equations of the form

    S:z=z(x,y),S':z'=z'(x',y'),                                (2)

and let a general ray R from the off-axis design focal pointF(x_(o),y_(o),z_(o)) pass through the lens surface S at the point x, y,zand through the surface S' at the point x',y'z'. At each of theinterfaces the ray is refracted in accordance with Snell's Law. The rayemerges from the lens in the direction ψ_(o), that is to say parallel tothe y,z-plane and at the angle ψ_(o) with respect to the z-axis. Let Wdenote the corresponding orthogonal wave front. In order to achieve adual ray structure from a symmetrically placed dual focal point at F'(x_(o),-y_(o),z_(o)) so that a general ray R' from the latter pointemerges from the lens in the corresponding dual direction -ψ_(o), werequire the lens surfaces S and S' to be symmetrical about the x,z-planeand, taking x_(o) =0, we require further and without loss of generality,that the surfaces S and S' also be symmetric about the y,z-plane.Finally, extremizing the lens design so as to obtain a maximal lensaperture, or what is the same thing, a minimal lens volume, by requiringthat the lens surfaces S and S' intersect in a sharp boundary curve, orlens periphery, Γ, our problem then is to so determine the shape of thelens and the form of the functions z=z(x,y) and z'=z'(x',y') specifyingthe lens surfaces S and S' in (2) as to insure that all rays R from thefocal point F(x_(o),y_(o),z_(o)) incident on the lens within the lensperiphery Γ are perfectly collimated by the lens into a parallel rayplane wave beam perfectly focused at infinity in the direction ψ_(o)and, by symmetry, that all rays from the dual focal pointF'(x_(o),-y_(o),z_(o)) similarly incident on the lens within theperiphery Γ are similarly perfectly focused by the lens at infinity inthe dual direction -ψ_(o).

Taking appropriate vector cross products between the incident ray R andthe normals to the lens surfaces S and S' at the points x,y,z, andx',y',z', respectively, in the illustration, and applying Snell's Law ateach interface, and using the Jacobian notation for appropriate two bytwo determinants of partial derivatives, we readily obtain for the lenssurfaces S and S' the system of partial differential equations ##EQU1##where A denotes the arguments A=(x,y,z,x',y',z') and the coefficientfunctions F(A) to G' (A) are of the forms ##EQU2## and there p and p'denote the path length elements defined by the expressions

    p.sup.2 =x.sup.2 +(y-y.sub.o).sup.2 +(z-z.sub.o).sup.2, p'.sup.2 =(x'-x).sup.2 +(y'-y).sup.2 +(z'-z).sup.2,                (5)

and are thus the geometrical lengths of the segments of the ray Rindicated in the figures.

In writing the partial differential equations (3) for the lens surfacesS and S' we have taken as independent variables the variables x and ywhich are the x,y coordinates of the point of intersection of the ray Rwith the lens surface S and we have considered the unknown coordinate zof this point and the unknown coordinates x',y',z' at which the ray Rintersects the lens surface S', each to be functions of these variables.

That is to say we suppose the equations (2) for S and S' can also bewritten in the form

S:z=z(x,y),S':z'=z'*(x,y), (6)

the latter of these being a parameterization of the equation for surfaceS' in terms of the variables x and y.

Essentially equivalent to either pair of the partial differentialequations in (4) and capable of being substituted for one of those pairsof equations by Malus's Theorem or, as may be shown by directsubstitution, is the optical path length condition that ##EQU3## for allrays R from the focal point F(x_(o),y_(o),z_(o)) to the wave front Wpropagating in the direction ψ_(o) with instantaneous intercept a on thez-axis as illustrated and, applying this relation to the extremal raysR_(o) and R'_(o) which pass through the lens extremities E(0,b_(o),0)and E'(0,-b_(o),0), we find for the focal point F(x_(o),y_(o),z_(o)) thecoordinates

    F:x.sub.o =0,y.sub.o =[b.sub.o.sup.2 +z.sub.o.sup.2 sec.sup.2 ψ.sub.o ].sup.1/2 sin ψ.sub.o,z.sub.o =-|z.sub.o |, (8)

and for the constant P_(o) in (7) the expression

    P.sub.o =[b.sub.o.sup.2 +z.sub.o.sup.2 sec.sup.2 ψ.sub.o ].sup.1/2 +a cos ψ.sub.o,                                          (9)

similarly as in the U.S. Pat. No. 3,146,451. In terms of (8) the dualfocal point is now F'(x_(o),-y_(o),z_(o)).

The natural symmetry of the lens surfaces S and S' about the x,z-planeimposed by the requirement of providing for a dual ray structure fromthe dual focus F'(x_(o),-y_(o),z_(o)) and the symmetry about thex,z-plane corresponding to the assumption that x_(o) =0 results in thefour symmetry conditions ##EQU4## as additional requirements on thefunctions defining the lens surfaces S and S' in (2) in terms of thevariables x,y and x',y' which in terms of the variables x and y asindependent variables become of course much more complicated relations.The symmetry conditions will subsequently be seen to play a very strongrole in the mathematics of the problem.

Applying the extremizing condition that the lens has a maximal apertureor, equivalently, has a minimal volume in the sense described previouslyso that the lens surfaces S and S' intersect in a sharp periphery Γpassing through the lens extremities E(0,b_(o),0) and E'(0,-b_(o),0) asin the figures, we readily derive as the boundary conditions of theproblem the requirement that that the functions z(x,y) and z'(x',y') in(2) satisfy the relations

    z(x,y)=z'(x,y)=0,                                          (11)

on the ellipse

    Γ:(x.sup.2 /b.sub.o.sup.2 cos.sup.2 ψ.sub.o)+(y.sup.2 /b.sub.o.sup.2)=1.                                        (12)

To derive (11) and (12) we use relations (7) to (10) all together andequate z(x,y) to z'(x,y).

Finally, similarly as in U.S. Pat. No. 3,146,451 we have as theconditions for F-number minimization that the exiting segment of the rayR_(o) from the focal point F(x_(o),y_(o),z_(o)) be tangent to the lenssurface S' at the lens extremity E(0,b_(o),0) and similarly for thecorresponding ray from the dual focal point F' (x_(o),-y_(o),z_(o)).Note that the lens in the illustration has an almost, but not quite,minimal F-number where F=|z_(o) |/2b_(o).

Our present invention is now characterized as a solution for the lenssurfaces S and S' in the boundary value problem consisting of the systemof partial differential equations (3) and (4) solved subject to theconditions (7) to (12) with or without the optional additional conditionof F-number minimization appended.

A variety of analytic, approximate, and numerical approaches andsolution algorithms are viable for application to the solution of thelens boundary value problem with nearly minimal F-numbers and, to theproblem with the F-number minimized. Such solutions were given by theinventor in U.S. Pat. No. 3,146,451 for the correspondingtwo-dimensional lens problem. Here we extended these solutions to thepresent partial differential equations problem in three dimensions.

To the foregoing end, ellipticizing the two-dimensional successiveapproximation solution of the lens problem presented in U.S. Pat. No.3,146,451 and appending appropriate Taylor series terms, we solve theboundary value problem (1) to (12) in the mixed forms ##EQU5## assumingthe F-number is nonminimal. Here the β_(n) (N) and β'_(n) (N) are thesuccessive approximation coefficients obtained in U.S. Pat. No.3,146,451 for the two-dimensional problem and β_(mn) and β'_(mn) arecorrection, or completion, terms corresponding to the slight but finitedeviations of the lens surfaces S and S' in the three-dimensionalproblem from a simple elliptic transformation of the corresponding lenssurfaces S and S' in the essentially two-dimensional problem discussedin U.S. Pat. No. 3,146,451.

To obtain the coefficients β_(n) (N) and β'_(n) (N) in the polynominalapproximations in (13) and (14) with the F-numbers nonminimal we beginexactly as in the two-dimensional problem of U.S. Pat. No. 3,146,451 byfitting 2N+2 rays R_(o),R'_(o),R_(N).sbsb.1,R'_(N).sbsb.1 . . . ,R_(N).sbsb.N and R'_(N).sbsb.N from the focal point F(x_(o),y_(o),z_(o))in the y,z-plane cross section of the lens as in FIG. 2 such that theserays are perfectly controlled in both direction and path length by thepolynominal approximates to the lens surfaces S and S' of degree 4N+2.As in U.S. Pat. No. 3,146,451 the technique hinges on solutions forapproximate values of the eigenvalue like angles φ₁ and φ'₁ in theillustration associated with the rays R_(N).sbsb.1 and R'_(N).sbsb.1which pass through the lens vertices, or axial intercepts of the lenssurfaces S and S', as shown. This solution in turn depends on solutionof simultaneous determinantal equations of the form ##EQU6## for thequantities φ₁ and φ'₁. In (15) and (16) the elements of the determinantsΔ(φ₁,φ'₁) and Δ' (φ₁, φ'₁) and, hence the determinants themselves,depend only on φ₁ and φ'₁ ; i.e., each element of each determinant iseither a constant or a function of φ₁ or φ'₁ or both. Moreover, thefunctions defining the elements are simple elementary algebraic andtrigonometric expressions as in U.S. Pat. No. 3,146,451 so no greatdifficulties attend the solution of the system (15) and (16) for φ₁ andφ'₁ for given N.

Letting the polynominal approximations in the solutions for the lenssurfaces S and S' in (13) converge to analytic limits and noting thatthe infinite series (13) are suitably convergent it then follows thatthe coefficients β_(mn) and β'_(mn) in the latter parts of (13) can beobtained by a Cauchy-Kovalevsky type process, i.e., by the method oflimits using known expressions for the derivatives (d^(k) /dy^(k))z(0,y)and (d^(k) /d'y^(k))z'(0,y') developed for the two-dimensional problemin U.S. Pat. No. 3,146,451 together with the chain rule, to compute thequantities ##EQU7## required in the process. For practical purposes onecan sometimes even determine a few of the coefficients by trial anderror using ray tracing methods to check the results. We use the lattermethod to obtain an approximate value of a single coefficient β'₁₁ inthe example worked out below.

Computational experience with the mixed forms of solution (13) and (14)indicates that the convergence of both the limit expressions and theinfinite series in (13) and (14) is exceedingly rapid even when theF-numbers are nearly minimal. Thus, for example, for the case of a lenswith F-number one-half at 20 degrees off-axis having as basic parametervalues the numbers

    |z.sub.o |=9,2b.sub.o =18,n.sub.o =1.594,ψ.sub.o =20; (19)

and λ_(o) arbitrary, and taking the simplest case N=1 in (13) and (14),as in FIG. 3 we obtain for the coefficients β_(n) (N) and β'_(n) (N) thevalues ##EQU8## which, with all β_(mn) and β'_(mn) values zero, yieldsthe ray tracing results shown in FIG. 4(a) and, which, with all of thelatter coefficients zero except for β'₁₁ which we find by trial anderror has approximately the value β'₁₁ =-1.5×10⁻⁵, yields the raytracing data shown in FIG. 4(b). In the first case the wave front isseen to be flat to within 0.01 units over a circular region of diameter18 cos ψ_(o) units, or to be flat to within one part in 1690, and in thesecond case it is seen to be flat to within 0.005 units over the samecircular region, or to be flat to within one part in 3380.

By the quarter wavelength Rayleigh criterion for diffraction limitedbeam forming, the latter case of the foregoing example lens with F=1/2at 20 degrees is capable of forming a collimated beam of half-powerwidth of approximately one-fifteenth of a degree, or four minutes ofarc, or twice this with a one-eighth wavelength Rayleigh criterion, avery sharply focused beam in either case for a bifocal lens with so verylow an F-number.

Finally, if the F-number is actually to be a minimum, the foregoingtreatment needs modification along the lines of the related well knowntreatment of the generalized aplanatic lens design problem as will beeasily recognized by those skilled in the art. In particular if such isto be the case, it will normally be best to give up the analyticexpressions for the lens surface S' in (13) and (14) and determine S'from S by satisfaction of the path length condition after previouslydetermining S as before.

After solving the lens boundary value problem we then seek to adjust thebasic lens parameters z_(o),n_(o),ψ_(o) in (1) particularly the index ofrefraction n_(o) and the off-axis design angle ψ_(o), so as to balanceastigmatism and optimize the lens for best balance of azimuth versuselevation scanning performance. Generally we seek a maximum azimuthscanning capability for given acceptable elevation performance. Incarrying out this azimuth versus elevation scanning trade off, any ofseveral varied but more or less equivalent optimization criteria may beapplied.

It has therefore been shown that an ellipticized azimuth versuselevation optimized and aperture extremized nonspherical lens havingperfect primary off-axis focal points F and F' and perfect conjugateoff-axis focal points F.sub.∞ and F'.sub.∞ at infinity is attainable. Ithas further been shown that such a lens balances the astigmatism in thesense of having negative astigmatism on axis, zero astigmatism at itsoff-axis or primary design foci F and F' and positive astigmatism atstill further off-axis points and can be optimized for best trade offbetween azimuth and elevation scanning capability by suitable optimalchoice at the basic lens parameter.

It will be understood that various changes in details, materials, stepsand arrangement of parts, which have been herein described andillustrated in order to explain the nature of the invention, may be madeby those skilled in the art within the principle and scope of theinvention as expressed in the appended claims.

What is claimed is:
 1. An ellipticized singlet azimuth versus elevationoptimized and aperture extremized nonspherical lens with surfaces S andS' specified by the partial differential equations ##EQU9## the symmetryconditions ##EQU10## and the boundary conditions

    z(x,y)=z'(x,y)=0,

on the ellipse

    Γ:(x.sup.2 /b.sub.o.sup.2 cos.sup.2 ψ.sub.o)+(y.sup.2 /b.sub.o.sup.2)=1

where S and S' are the lens surfaces having functional representationsof the forms z=z(x,y) and z'=z'(x',y') wherein x' and y' are themselvesfunctions x'=x'(x,y) and y'=y'(x,y) of the independent variables x andy, ∂z/∂x and ∂z/∂y are the partial derivatives of z with respect to theindependent variables x and y respectively, ##EQU11## are the Jacobianof z' and y' with respect to the independent variables x and y, theJacobian of x' and y' with respect to the independent variables x and yand the Jacobian of x' and z' with respect to the independent variablesx and y respectively, F(A), G(A), F'(A) and G'(A) are the functions ofthe arguments A=(x,y,z,x',y',z') defined as ##EQU12## and where p and p'denote the path length elements defined by the expressions

    p.sup.2 =x.sup.2 +(y-y.sub.o).sup.2 +(z-z.sub.o).sup.2,p'.sup.2 =(x'-x).sup.2 +(y'-y).sup.2 +(z'-z).sup.2,

and n_(o), y_(o), z_(o) and ψ_(o) are respectively, the index ofrefraction of the lens material, the y and z coordinates of the finitefocal point F and the off-axis angle to the infinite focal pointF.sub.∞, Γ is the ellipse bounding the lens formed by S and S' and isdefined by the equation shown in which x and y are the independentvariables and b_(o) is the maximum radius of the lens formed by S and S'and the semi-major axis of the ellipse Γ.
 2. A lens according to claim 1wherein said lens is a microwave lens.
 3. An ellipticized lens accordingto claim 1 wherein said lens has two perfect off-axis focal points F andF' at a finite distance, and two perfect off-axis focal points F.sub.∞and F'.sub.∞ at an infinite distance, said focal points being in thedirections ±ψ_(o) with respect to the lens axis.
 4. A lens according toclaim 3 wherein said lens is a microwave lens.
 5. An ellipticized lensaccording to claim 1 wherein said lens has two perfect off-axis finitefocal points F and F' forming perfectly collimated plane wave beamsdirected toward infinity in the two off-axis directions ±ψ_(o).
 6. Alens according to claim 5 wherein said lens is a microwave lens.
 7. Anellipticized lens according to claim 1 wherein said lens has balancedastigmatism having negative astigmatism on axis, zero astigmatism at thetwo primary focal points F and F' and positive astigmatism at faroff-axis focal points.
 8. A lens according to claim 7 wherein said lensis a microwave lens.
 9. An ellipticized lens according to claim 1wherein said lens has wide angle scanning capability providing increasedscan in one plane at a sacrifice in scan in an orthogonal plane.
 10. Alens according to claim 9 wherein said lens is a microwave lens.
 11. Anellipticized lens according to claim 1 wherein said lens has maximalaperture and minimal volume.
 12. A lens according to claim 11 whereinsaid lens is a microwave lens.
 13. An ellipticized lens according toclaim 1 wherein said lens is of substantially minimal F-number.
 14. Alens according to claim 13 wherein said lens is a microwave lens.